Gödel's second incompleteness theorem states that, under the effectiveness and arithmetic-strength hypotheses of Gödel's
first incompleteness theorem, and with the usual arithmetical encoding of proofs, the proposition 
formalizing the consistency
of a consistent axiomatic
system 
has no proof in 
. Every proposition has a proof in an inconsistent axiomatic
system, so the consistency assumption is essential.
Gödel's Second Incompleteness Theorem
See also
Consistency, Gödel's Completeness Theorem, Gödel's First Incompleteness Theorem, Gödel's Incompleteness Theorems, Peano ArithmeticExplore with Wolfram|Alpha
References
Gödel, K. "Über Formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme, I." Monatshefte für Math. u. Physik 38, 173-198, 1931. https://doi.org/10.1007/BF01700692.Gödel, K. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. New York: Dover, 1992.Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 17, 1989.Rucker, R. Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1995.Referenced on Wolfram|Alpha
Gödel's Second Incompleteness TheoremCite this as:
Weisstein, Eric W. "Gödel's Second Incompleteness Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GoedelsSecondIncompletenessTheorem.html